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Theorem rspceaov 27385
Description: A frequently used special case of rspc2ev 2968 for operation values, analogous to rspceov 5977. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
rspceaov  |-  ( ( C  e.  A  /\  D  e.  B  /\  S  = (( C F D))  )  ->  E. x  e.  A  E. y  e.  B  S  = (( x F y))  )
Distinct variable groups:    x, A    x, y, B    x, C, y    y, D    x, F, y    x, S, y
Allowed substitution hints:    A( y)    D( x)

Proof of Theorem rspceaov
StepHypRef Expression
1 eqidd 2359 . . . 4  |-  ( x  =  C  ->  F  =  F )
2 id 19 . . . 4  |-  ( x  =  C  ->  x  =  C )
3 eqidd 2359 . . . 4  |-  ( x  =  C  ->  y  =  y )
41, 2, 3aoveq123d 27366 . . 3  |-  ( x  =  C  -> (( x F y))  = (( C F y))  )
54eqeq2d 2369 . 2  |-  ( x  =  C  ->  ( S  = (( x F
y)) 
<->  S  = (( C F y))  ) )
6 eqidd 2359 . . . 4  |-  ( y  =  D  ->  F  =  F )
7 eqidd 2359 . . . 4  |-  ( y  =  D  ->  C  =  C )
8 id 19 . . . 4  |-  ( y  =  D  ->  y  =  D )
96, 7, 8aoveq123d 27366 . . 3  |-  ( y  =  D  -> (( C F y))  = (( C F D))  )
109eqeq2d 2369 . 2  |-  ( y  =  D  ->  ( S  = (( C F
y)) 
<->  S  = (( C F D))  ) )
115, 10rspc2ev 2968 1  |-  ( ( C  e.  A  /\  D  e.  B  /\  S  = (( C F D))  )  ->  E. x  e.  A  E. y  e.  B  S  = (( x F y))  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1642    e. wcel 1710   E.wrex 2620   ((caov 27296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-res 4780  df-iota 5298  df-fun 5336  df-fv 5342  df-dfat 27297  df-afv 27298  df-aov 27299
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